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Time-Dependent Volatilities Multi-Stage Compound Real Option

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Time-Dependent Volatilities Multi-Stage Compound Real Option Time-dependent Volatility Multi-stage Compound Real Option Model and Application∗ Pu Gong∗∗, Zhi-Wei He, Jian-Ling Meng School of Management Huazhong University of Science and Technology Wuhan, P.R.China, 430074 Abstract The simple compound option model has many limitations when applied in practice. The research on compound option theory mainly focuses on two directions. One is the extension from two-stage to multi-stage, and the other is the modification of the stochastic difference equations which describe the movement of underlying asset value. This paper extends the simple compound option model in both two directions and proposes the Time-dependent Volatility Multi-stage Compound Real Option Model. Due to the introduction of time-dependent volatility, it is difficult to derive the closed-form solution by the traditional analytical approach. This paper presents the pricing governing partial differential equation, proposes the boundary conditions and terminal conditions, and then gets the numerical solution by Finite Differential Methods. Finally this paper applies Time-dependent Volatility Multi-stage Compound Real Option Model to evaluate venture capital investment. The numerical results show that the Fixed Volatility Multi-stage Compound Real Option Model underestimates significantly the intrinsic value of venture capital investment as well as exercise threshold of later stages, but overestimates the exercise threshold of earlier stages. Keywords: Time-dependent volatility, Multi-stage compound real option, Contingent Claim Analysis, Finite Differential Method, Venture capital investment 1 INTRODUCTION Compound option is option on option[1]. Research about compound option model began with the path breaking work on option pricing of Black and Scholes. They view the stock as option on the firm value, and if the firm value can be viewed as the option of the liability of firm, then the stock can be modelled as the compound option on liability of firm. Since Geske derived the closed form solution for the simple two-stage compound European option model[2][3], compound option model has been widely used. In essence, compound option model reflects one right sequence which compounds each other. So it is suitable to describe problems involved sequential decision. Generally R&D projects have multi-stage nature[4]. Only when the research or management goal of earlier stage is achieved, project can enter the next stage. Venture capital investment is another typical multi-stage investment[5]. If the given goal during the operation is not achieved, the venture capitalist can cancel the investment of next stage. Another multi-stage decision example is the firm strategy decision[6]. In fact, when the management make the strategy, they are not only interested in the direct predictable cash flow, but also the potential future investment opportunities ∗ This research is supported by National Natural Science Foundation of China (70471043). ∗∗ Corresponding author. Tel.: +86-27-8755-6489; Fax: +86-27-8755-6437; E-mail addresses: [email protected] 1 accompanying the current investment, which generate considerable cash flow in the future or make the firm stay in favour competition position. The above three examples are full of management flexibility and strategy flexibility. Such multi-stage decision problems can be modelled as compound option. The simple compound option model is based on the Black-Scholes framework. The assumption that the underlying asset value follows Geometric Brown Motion is very strict. Furthermore, the two-stage setting seems too simple. In practice, the simple compound option model has many limitations. The current research about compound option model mainly focus on the development of simple compound option model in two directions. One is the extension from two-stage to multi-stage. Dixit and Pindyck make use of multi-stage compound option to model the sequential investment[7]. They adopt Dynamic Programming and Contingent Claims Analysis (CCA) to derive the governing partial differential equation(PDE), and then give the analytical solution for value function and exercise threshold of compound option under some specific boundary conditions. However, numerical methods are required to solve such a two dimensions parabolic PDE under most conditions. Alvarez and Stenbacka develop a mathematical approach based on the Green representation of Markovian functionals to find the value function of compound option and the optimal exercise rules[8]. Lin extends directly the conclusion of simple compound option to multi-stage compound real option, presented the closed form solution, and compared several numerical calculating method for the solution[5]. The main shortcoming lies in the existence of nested high-dimension integral. Since the computing complexity and cost increase rapidly with the total stage number, the traditional analytical approach is hardly used, and it is difficult to derive the closed form solution. The calculation costs large computing resource. The other research direction is the improvement of stochastic differential equations which describe the movement of the underlying asset value. In the simple situation, model only consider single underlying asset, and assume that the movement of underlying asset value can be modelled as Geometric Brownian Motion. However, in the multi-stage situation, the sensitivity of the underlying motion parameters is amplified and then the simple assumption of fixed volatility and fixed return rates appears more unpractical. Buraschi and Dumas derive a solution for the valuation of compound options when the underlying asset value follows a general diffusion process[9]. The solution can be expressed as a forward integral of the price surface of European plain vanilla options. Geman, El Karoui[10] and Rochet, and Elettra and Rossella[11] focus on relaxation of the Geometric Brownian Motion assumption and introduce the time-dependent volatility. In their model they also consider the time-dependent interest rate and extend the model into two underlying asset situation. They derive the analytical solution for two-stage European compound option. However the solution still includes high-dimension integrals. Herath and Park extend the binomial lattice framework to model a multi-stage investment as a compound real option on several uncorrelated underlying variables[12]. However, they don’t consider the accuracy and convergence of the numerical solution, and the calculation of the exercise threshold. Only in few specific situations we can derive the analytical solution for compound option model. Many researcher adopted modern numerical technique to find the solution. Trigeorgis presents a numerical method called Log-Transformed Binomial Numerical Analysis Method[13], to value complex investments with multiple interacting options, including compound option. The method can achieve good consistency, stability, and efficiency. Breen presents Accelerated Binomial Option Pricing Model based on the binomial and Geske-Johnson models[14], which is faster than 2 the conventional binomial model and applicable to a wide range of option pricing problems. Though the numerical methods based on Binomial Option Pricing Model is easy to use, but in multi-stage situation the convergence of numerical solution is very slow and the computing cost will be very large. In fact, there are many other available numerical techniques, such as Finite Differential Method (FDM), Finite Elements Method, and etc, which are applicable for more complex problems and achieve better accuracy, convergence and stability. In this paper we develop the simple compound option both in two directions. We introduce time-dependent volatility into the multi-stage compound real option model, based on the Fixed Volatility Multi-stage Compound Real Option Model (FV-MCROM) presented by Lin[5]. Because of the introduction of the time-dependent volatility, it is hard to derive the analytical solution for the Time-dependent Volatility Multi-stage Compound Real Option Model (TV-MCROM). We use CCA to establish the governing PDE, and then propose the boundary conditions and terminal conditions. But in this paper we don’t try to derive the analytical solution, but apply FDM to find the numerical solution. Finally we present one application of the TV-MCROM in venture capital investment evaluation, in which we can conclude that such a development does really make sense. In the next section we briefly review the FV-MCROM. And section 3 proposes the TV-MCROM and presents the governing PDE, terminal conditions and boundary conditions. We detail the solving procedure of PDE by FDM in section 4. In Section 5 we apply the TV-MCROM to evaluate the venture capital investment with. Section 6 concludes. 2 REVIEWS OF THE FIXED VOLATILITY MULTI-STAGE COMPOUND REAL OPTION MODEL The right sequence of investor embedded in a multi-stage project can be viewed as a series of real option. Its value consists of two components: one is value of current direct cash flow; the other is value of following investment right. In each stage the investor receives the cash flow of that stage and at the same time decides whether he will exercise the real option or not. If exercising, the investor will purchase the real option, i.e. investment right, of next stage at a certain exercise price, i.e. investment outlay, thus the project continues. If not, the investor keeps the investment outlay and abandons the following investment right, thus the project is abandoned. The investor
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